I have a context free grammar (CFG). The initial symbol is `A1`. The terminal symbols are just integers and the non-terminal symbols start with `A`. I also have a target sequence of terminal symbols which needs to be produced by the grammar. However, the CFG production rules have had some terminal symbols removed which may now render it impossible to produce the target sequence. Please attempt to repair the CFG so that the sequence can be produced. The only operations which can be made to repair the CFG are the insertion of terminal symbols anywhere in existing production rules. Deleting or reordering symbols in existing production rules and creating new production rules is prohibited. Please try minimize the total number of terminal symbols inserted into the production rules so that the CFG can produce the target sequence. Reason about the problem however you wish and write out whatever working out you want. When you are finished please write a new line with `Updated production rules:`, and then write a new line for each of the updated production rules (including `rule_<n>:`), no need to rewrite the production rules which haven't changed. Do not write anything after the updated production rules.


"""
There are 39 terminals, ranging from 1 to 39.
Each terminal symbol occurs in the target sequence at least once.

There are 80 non-terminal symbols, ranging from A1 to A80.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> A79 36 A80 A78
rule_3: A3 -> A4 A5
rule_4: A4 -> 7
rule_5: A5 -> 5 A6
rule_6: A5 -> 6 A7
rule_7: A6 -> 1
rule_8: A6 -> 2
rule_9: A6 -> 3
rule_10: A6 -> 4
rule_11: A7 -> 1
rule_12: A7 -> 2
rule_13: A7 -> 3
rule_14: A7 -> 4
rule_15: A8 -> A9 A10
rule_16: A9 -> 8
rule_17: A10 -> 5 A11
rule_18: A10 -> 6 A12
rule_19: A11 -> 1
rule_20: A11 -> 2
rule_21: A11 -> 3
rule_22: A11 -> 4
rule_23: A12 -> 1
rule_24: A12 -> 2
rule_25: A12 -> 3
rule_26: A12 -> 4
rule_27: A13 -> A14 A15
rule_28: A14 -> 15
rule_29: A15 -> 13 A16
rule_30: A15 -> 14 A17
rule_31: A16 -> 1
rule_32: A16 -> 2
rule_33: A16 -> 3
rule_34: A16 -> 4
rule_35: A17 -> 1
rule_36: A17 -> 2
rule_37: A17 -> 3
rule_38: A17 -> 4
rule_39: A18 -> A19 A20
rule_40: A19 -> 16
rule_41: A20 -> 13 A21
rule_42: A20 -> 14 A22
rule_43: A21 -> 1
rule_44: A21 -> 2
rule_45: A21 -> 3
rule_46: A21 -> 4
rule_47: A21 -> 17
rule_48: A22 -> 1
rule_49: A22 -> 2
rule_50: A22 -> 3
rule_51: A22 -> 4
rule_52: A23 -> A24 A25
rule_53: A24 -> 18
rule_54: A25 -> 20 A26
rule_55: A25 -> 21 A27
rule_56: A26 -> 1
rule_57: A26 -> 2
rule_58: A26 -> 3
rule_59: A26 -> 4
rule_60: A26 -> 19
rule_61: A27 -> 1
rule_62: A27 -> 2
rule_63: A27 -> 3
rule_64: A27 -> 4
rule_65: A28 -> A29 A30
rule_66: A29 -> 22
rule_67: A30 -> 20 A31
rule_68: A30 -> 21 A32
rule_69: A31 -> 1
rule_70: A31 -> 2
rule_71: A31 -> 3
rule_72: A31 -> 4
rule_73: A32 -> 1
rule_74: A32 -> 2
rule_75: A32 -> 3
rule_76: A32 -> 4
rule_77: A33 -> 10 A34
rule_78: A33 -> 11 A35
rule_79: A34 -> 1
rule_80: A34 -> 2
rule_81: A34 -> 3
rule_82: A34 -> 4
rule_83: A34 -> 23
rule_84: A35 -> 1
rule_85: A35 -> 2
rule_86: A35 -> 3
rule_87: A35 -> 4
rule_88: A36 -> 10 A37
rule_89: A36 -> 11 A38
rule_90: A37 -> 1
rule_91: A37 -> 2
rule_92: A37 -> 3
rule_93: A37 -> 4
rule_94: A38 -> 1
rule_95: A38 -> 2
rule_96: A38 -> 3
rule_97: A38 -> 4
rule_98: A38 -> 24
rule_99: A39 -> A40 A36
rule_100: A40 -> 25
rule_101: A41 -> A42 A33
rule_102: A42 -> 26
rule_103: A43 -> 27 A41
rule_104: A43 -> A44
rule_105: A43 -> A45
rule_106: A43 -> A46
rule_107: A43 -> A47
rule_108: A43 -> A48
rule_109: A43 -> A49
rule_110: A43 -> A50
rule_111: A43 -> A51
rule_112: A43 -> A52
rule_113: A43 -> A53
rule_114: A43 -> A54
rule_115: A43 -> A55
rule_116: A44 -> 9
rule_117: A45 -> 9
rule_118: A46 -> 9
rule_119: A47 -> 9
rule_120: A48 -> 9
rule_121: A49 -> 9
rule_122: A50 -> 9
rule_123: A51 -> 9
rule_124: A52 -> 9
rule_125: A53 -> 9
rule_126: A54 -> 9
rule_127: A55 -> 9
rule_128: A56 -> 29 A39
rule_129: A57 -> A58
rule_130: A57 -> A59
rule_131: A57 -> A60
rule_132: A57 -> A61
rule_133: A57 -> A62
rule_134: A57 -> A63
rule_135: A57 -> A64
rule_136: A57 -> A65
rule_137: A57 -> A66
rule_138: A57 -> A67
rule_139: A57 -> A68
rule_140: A57 -> A69
rule_141: A58 -> 9
rule_142: A59 -> 9
rule_143: A60 -> 9
rule_144: A61 -> 9
rule_145: A62 -> 9
rule_146: A63 -> 9
rule_147: A64 -> 9
rule_148: A65 -> 9
rule_149: A66 -> 9
rule_150: A67 -> 9
rule_151: A68 -> 9
rule_152: A69 -> 9
rule_153: A70 -> 31 A71
rule_154: A71 -> 12
rule_155: A72 -> A73 A70 A3 A57
rule_156: A73 -> 32 A8
rule_157: A74 -> A75 A56 A23 A43
rule_158: A75 -> 33 A28
rule_159: A76 -> 35
rule_160: A77 -> 35
rule_161: A78 -> A74 34
rule_162: A79 -> A72 A76
rule_163: A79 -> A74 A77
rule_164: A80 -> 37 A13 A13
rule_165: A80 -> 37 A13 A18 38

Target sequence:
32 8 6 1 31 12 7 5 2 9 35 36 37 15 14 3 39 16 13 17 38 33 22 21 4 29 25 11 24 30 18 20 19 27 26 10 23 28 34
"""